The main results for the two-orbital Hubbard model at half-filling for the parent compound are shown in Fig. REF . Fig. REF b and REF c show the charge density profile, defined as \(n(x)=\sum _{y=1}^{L_y}\langle \hat{n}^{[l]}_i\rangle /L_y\) , for the Ni and \(R\)  layers, respectively. Here \(\hat{n}^{[l]}_i\)  is the electron number operator on site \(i=(x,y)\)  where \(x\)  and \(y\)  are the rung and row indices of the \(l\)  (\(l=\)  Ni, \(R\) ) layer. It is clear that a small fraction of electrons transfer from the Ni layer to the \(R\)  layer, which gives rise to ‘self-doped’ holes in the Ni layer. The electron densities in the \(R\)  layer are approximately \(6.87\%\)  for \(\varepsilon _0^{R\text{-Ni}}=3.7\)  and \(10.04\%\)  for \(\varepsilon _0^{R\text{-Ni}}=3.4\) , respectively, where \(\varepsilon _0^{R\text{-Ni}}\)  is the on-site energy difference between the \(R\)  and the Ni orbital. The electron densities are consistent, in the range of \(\varepsilon _0^{R\text{-Ni}}\)  used in our simulation, with the Hall coefficient measurement on the size of the rare-earth electron pocket, where there are approximately \(8\%\)  electrons per formula unit for NdNiO\(_2\)  [1]}. Although the non-interacting \(R\)  band has much higher energy than the Ni band (see supplementary material [2]} for more details), strong on-site Hubbard \(U_{\text{Ni}}\)  splits the Ni band into upper and lower Hubbard bands, broadening the interacting Ni bandwidth significantly.
